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Congruent Triangles Test Answer Key with Scoring – Geometry

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Last updated: October 22, 2026 · Written by the Math Made Easy math teachers

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Looking for a congruent triangles test answer key? Then this answer key gives the solutions and the scoring guide of the Geometry test on Proving Triangles Congruent: SSS to HL. Each solution gives the reasons and the points awarded. A proof earns partial credit for correct congruent pairs even if the final criterion is missing.

Standards: G-CO.B.6, G-CO.B.7, G-CO.B.8, G-SRT.B.5 · Level: Grade 10 Geometry

Question 1 Corresponding parts with algebra (2 points)

  1. 4x − 3 = 2x + 9 gives x = 6, so DE = 2(6) + 9 = 21.
  2. 7y − 6 = 4y + 18 gives 3y = 24 and y = 8, so m∠F = 4(8) + 18 = 50°.

(a) DE = 21, (b) 50°

Scoring: 1 point per part.

Question 2 Choose the criterion (3 points)

  1. ∠B lies between AB and BC: SAS.
  2. AC lies between ∠A and ∠C: ASA.
  3. The angle is not between the two sides: SSA, so none.

(a) SAS, (b) ASA, (c) none

Scoring: 1 point per part.

Question 3 Right triangles (3 points)

  1. Hypotenuse and a leg of right triangles: HL.
  2. BC = √(15² − 9²) = √144 = 12.
  3. Area = ½ · 9 · 12 = 54, using the congruent leg EF = 12.

(a) HL, (b) BC = 12, (c) 54 square units

Scoring: 1 point per part.

Question 4 Prove with SAS (3 points)

ABCD
Solved figure with the alternate interior angles and the shared diagonal marked.
Statement Reason
1. AB ∥ DC and AB ≅ DC Given
2. ∠BAC ≅ ∠DCA Alternate interior angles (parallel lines)
3. AC ≅ CA Reflexive Property
4. △ABC ≅ △CDA SAS

Scoring: 1 point for step 2 with reason, 1 point for step 3, 1 point for the SAS conclusion.

Question 5 Congruence on the plane (3 points)

△RST: RS = 6, ST = 8, RT = √(6² + 8²) = 10.

△UVW: UV = 7 − 1 = 6, VW = 9 − 1 = 8, UW = √(8² + 6²) = 10.

The side lengths 6, 8, 10 match, so △RST ≅ △UVW by SSS.

Congruent by SSS

Scoring: 1 point for each triangle’s side lengths, 1 point for the SSS conclusion.

Question 6 Why SSA fails (2 points)

No. The given parts form an SSA pattern. The side of length 7 can meet the base ray in two different places, producing two different triangles with the same three measurements.

No, SSA does not prove congruence

Scoring: 1 point for “no,” 1 point for a valid explanation.

Question 7 Truss beams (2 points)

DF ≅ CA, so DF = 5.6 m. One triangle uses 3.5 + 4.2 + 5.6 = 13.3 m of beam; two use 26.6 m. Cost = 26.6 × 12.50 = $332.50.

DF = 5.6 m; cost $332.50

Scoring: 1 point for DF, 1 point for the cost.

Question 8 Paragraph proof with ASA (2 points)

Because BD bisects ∠ABC, ∠ABD ≅ ∠CBD. Because BD bisects ∠ADC, ∠ADB ≅ ∠CDB. The side BD is shared, so BD ≅ BD by the Reflexive Property. The shared side lies between the congruent angles, so △ABD ≅ △CBD by ASA. Then AB ≅ CB by CPCTC.

AB ≅ CB

Scoring: 1 point for the ASA argument with all three pairs, 1 point for the CPCTC conclusion.

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